In Paul Selick's book Introduction to Homotopy Theory, he says that one can prove the CW-Approximation Theorem:
Given a topological space $Y$ there exists CW-complex $X$ and a map $f : X \rightarrow Y$ such that $f_\ast : \pi_n(X) \rightarrow \pi_n(Y)$ is an isomorphism for all $n$.
using the fact that
Let $f : X \rightarrow Y$ be a map between CW-complex. Then there exists $g : X \rightarrow Y$ such that $g \simeq f$ and $g$ is cellular.
However he does not give any detail and I don't see how one should proceed.
In Hatcher in his book available here : http://www.math.cornell.edu/~hatcher/AT/AT.pdf describes "cellular approximation" that results in theorem 4.8 page 349, which is what you want to prove.
As usual with CW-complexes, this is prove by induction on the $n$-dimensional skeletons. The details can be quite complex depending on what you already know true about CW-complexes or not, I'm letting you read the text and ask another question if you don't understand.